How to determine if the curve is closed

Question

There is a given curve $ α(t) = (3 \cos t − \cos 3t, 3 \sin t − \sin 3t)$ and I have to determine if it's a closed one. I tried to find the answer and failed. What should I do? Thanks.

Answer 1

Since this curve is periodic with a period $2\pi$, and it doesn't have any pole (any value of $t$ makes $\alpha(t)$ undefined), so after every $2\pi$ this curve will come back to the same point, and going to loop and loop with the same period infinitely, this curve is definitely closed. Hope this help!

Answer 2

With $\alpha(t)= (3\cos t−\cos 3t, 3\sin t−\sin 3t)$ we see $$\alpha(0)=(2,0)=\alpha(2\pi)$$ shows this path is closed in $[0,2\pi]$. Indeed since $\sin$ and $\cos$ are periodic, then $3\cos t−\cos 3t$ and $3\sin t−\sin 3t$ are also peridic with period $2\pi$ then in every interval $[t_0,t_0+2\pi]$ we have $\alpha(t_0)=\alpha(t_0+2\pi)$.